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What is the precise relation between Witt vectors and Witt polynomials? Thanks. Jakob.scholbach 20:27, 3 April 2007 (UTC)
Isn't there some finite-dimensional algebra of Witt n-vectors, of which this is presumably the limit as n goes to infinity? That's the impression I got from looking at Serre's article "Sur la topologie des varietes algebriques en caracteristique p."
What's confusing is that the allegedly different Teichmüller representatives, if defined as the solutions of xp − x = 0 in Fp, are just Fp. And if we take {0, 1, 2, ..., p − 1} as the representatives of Fp, then nothing has changed! What's really going on is that we want the solutions of xp − x = 0 in Zp, and each such solution mod p gives a distinct element in Fp. We can then pick our usual representatives for Fp to represent the Teichmüller representatives, but the addition and multiplication arithmetic would be different, as the rest of the article describes. 128.105.2.11 ( talk) 21:10, 9 June 2011 (UTC)
I just re-wrote the entire section called "details", hopefully making it much more clear as to what is going on. I did not change the flow or development of that section, but sharpened the focus, tightened up the language, removed much of the ambiguity. I had found it painfully impossible to understand as originally written; I hope its OK now. Well, the last half of the argument could use some more cleanup, but I'm exhausted for now. 67.198.37.16 ( talk) 19:49, 5 July 2016 (UTC)
The sentence "The ghost components can be thought of as an alternative coordinate system ..." strikes me as deeply misleading, since when the ring you start with has characteristic p as it usually will, the map to ghost components is definitely not injective.
Let R be perfect of characteristic p. Since has nilpotent kernel, deformation theory tells us that the category of formally étale algebras over and are equivalent, and the functor is given by nothing but . Furthermore, the Witt vector is the unique p-adically complete p-torsionfree -algebra lifting R. -- Fourier-Deligne Transgirl ( talk) 08:49, 17 November 2022 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
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What is the precise relation between Witt vectors and Witt polynomials? Thanks. Jakob.scholbach 20:27, 3 April 2007 (UTC)
Isn't there some finite-dimensional algebra of Witt n-vectors, of which this is presumably the limit as n goes to infinity? That's the impression I got from looking at Serre's article "Sur la topologie des varietes algebriques en caracteristique p."
What's confusing is that the allegedly different Teichmüller representatives, if defined as the solutions of xp − x = 0 in Fp, are just Fp. And if we take {0, 1, 2, ..., p − 1} as the representatives of Fp, then nothing has changed! What's really going on is that we want the solutions of xp − x = 0 in Zp, and each such solution mod p gives a distinct element in Fp. We can then pick our usual representatives for Fp to represent the Teichmüller representatives, but the addition and multiplication arithmetic would be different, as the rest of the article describes. 128.105.2.11 ( talk) 21:10, 9 June 2011 (UTC)
I just re-wrote the entire section called "details", hopefully making it much more clear as to what is going on. I did not change the flow or development of that section, but sharpened the focus, tightened up the language, removed much of the ambiguity. I had found it painfully impossible to understand as originally written; I hope its OK now. Well, the last half of the argument could use some more cleanup, but I'm exhausted for now. 67.198.37.16 ( talk) 19:49, 5 July 2016 (UTC)
The sentence "The ghost components can be thought of as an alternative coordinate system ..." strikes me as deeply misleading, since when the ring you start with has characteristic p as it usually will, the map to ghost components is definitely not injective.
Let R be perfect of characteristic p. Since has nilpotent kernel, deformation theory tells us that the category of formally étale algebras over and are equivalent, and the functor is given by nothing but . Furthermore, the Witt vector is the unique p-adically complete p-torsionfree -algebra lifting R. -- Fourier-Deligne Transgirl ( talk) 08:49, 17 November 2022 (UTC)